The proliferation of LLM-based conversational agents has resulted in excessive disclosure of identifiable or sensitive information. However, existing technologies fail to offer perceptible control or account for users' personal preferences about privacy-utility tradeoffs due to the lack of user involvement. To bridge this gap, we designed, built, and evaluated Rescriber, a browser extension that supports user-led data minimization in LLM-based conversational agents by helping users detect and sanitize personal information in their prompts. Our studies (N=12) showed that Rescriber helped users reduce unnecessary disclosure and addressed their privacy concerns. Users' subjective perceptions of the system powered by Llama3-8B were on par with that by GPT-4o. The comprehensiveness and consistency of the detection and sanitization emerge as essential factors that affect users' trust and perceived protection. Our findings confirm the viability of smaller-LLM-powered, user-facing, on-device privacy controls, presenting a promising approach to address the privacy and trust challenges of AI.